Not necessarily.
Null and sensitivity are generally better in larger loops, but
the nulls are principally determined by the capacitive balance
of the loop.

Is
the directionality influenced by the distance between windings?

Not necessarily.
A lot has to do with the physical construction. The term "balanced"
(physically and electronically) has some meaning here.

Will
the sensitivity be improved by taking more windings or a larger
frame? (Capacitor should of course be reduced to keep LC product
in the right range.)

Yes, either
will improve sensitivity. The output voltage is directly proportional
to the loop area and the number of turns.

Is
it true that the low directionality for some stations is due
to electrical pickup?

Yes; imbalances
in the loop (principally capacitive coupling) cause the entire
loop structure to act as a short vertical antenna which is added
to the loop output. This is often called "antenna effect".

Re-radiation
from electrical wires, appliances, etc. can also cause "skewing"
of receiving patterns and nulls. In such cases, the expected signal
may be best, or the QRM nulled best, at unexpected directions.
I've found the best place to use a loop antenna is in a middle
of a field where it's sometimes just as easy to erect a Beverage
antenna...

Where
does the antenna take most of this pickup: in the primary coil
(28m wire), the secondary coil (8m), or the feed line (2m of
twin-lead chord (perhaps coax would be better?)?

If the loop
is well-balanced (principally symmetrical capactance to ground
from all parts of the loop, including the pickup and feedline),
then most of the pickup should come from the tuned main loop (28m
wire).

Instead of
the 2m twin-lead chord you should feed the signal to the receiver
with coaxial cable. Otherwise this is very possibly a secondary
pickup point of signals (especially local ones), giving you inconsistent
sensitivity and null.

How
can electrical pickup be reduced? I tried grounding the primary
coil but this kills the signal. Grounding the radio or the secondary
coil has almost no effect on interference or directionality.

´ No
mystery there. Shunting the signal to ground will kill it. The
ability to tilt the loop (from side to side) is also important
to attain deep nulls and to reduce local noise. Antenna phasing
units can sometimes also be used to cancel noise. I have used
my own phasing unit for this purpose. Several such products are
available commercially, one of them from from JPS another from
S.E.M.